Julia Set as a Martin Boundary

Abstract

The Julia set of the class of hyperbolic rational maps having a totally disconnected Julia set is here identified as the Martin boundary of a Markov chain by using symbolic dynamics. When the Julia set is also bounded, this connection allows one to relate various thermodynamic concepts, such as entropy, measure of maximal entropy, Gibbs measure, and measure of equilibrium, to potential theoretic concepts such as capacity and harmonic measures on the Julia set on which a suitable potential function is defined. We have identified the measure of maximal entropy for that class of rational maps on the Julia set; it is simply the image measure of a certain Bernoulli measure on the shift space. We have also proven that the harmonic measure on the Julia set is the image measure of a non-atomic, quasi-invariant, conservative measure on the one-sided shift space.We have further shown that this quasi-invariant measure is a Gibbs measure and is equivalent to a Bernoulli measure. By using the Ruelle-Perron-Frobenius theorem we have deduced that the Gibbs measure gives rise to a unique, shift-invariant equilibrium measure. We have further established that the measure of equilibrium for the logarithmic potential and the classical harmonic measure coincide in our case (with the Julia set being bounded and totally disconnected). Finally, we have proven that a certain Dirichlet problem for the Fatou domain in our case has a unique solution.